Question

Approximate √19 to the nearest hundredth

Answer 1

You could use the Babylonian method to approximate the square root of 19.

Pick a starting guess, such as \(\sqrt{16} = 4\). Let that be \(x_0\). To make your next guess, let \[x_1 = \frac{1}{2}(x_0 + \frac{19}{x_0})\]Now do the same step again, except substitute \(x_1\) where you have \(x_0\), and find the next approximation, \(x_2\). Repeat until the result stays the same through the desired decimal place.

Answer 2

Is there another way to do it?

Answer 3

Sure, you can get out your calculator and start multiplying 4*4, 4.1*4.1, 4.2*4.2, etc. and zeroing in on it that way. Or you could use the Newton-Raphson method, or try a linear approximation from a nearby point where you know the square root, or try to guess at it by using \[(a+b)^2 = a^2 + 2ab + b^2\] with \(a=4\) and \(b = \) some small value, or ...

Babylonian method is probably the easiest, and you'll only need to do it 2 or 3 times.

Answer 4

thank you, i'll try it!